On the Aizenman exponent in critical percolation

The probabilities that clusters span a hypercube of dimensions two to seven along one axis of a percolation system under criticality were investigated numerically. We used a modified Hoshen-Kopelman algorithm combined with Grassberger's «go with the winner» strategy for the site percolation. We performed a finite-size analysis of the data and found that the probabilities confirm Aizenman's proposal for the multiplicity exponent for dimensions three to five. A crossover to the mean-field behavior around the upper critical dimension is also discussed.